US8781937B2 - Optimal portfolio withdrawal during retirement in the presence of longevity risk - Google Patents
Optimal portfolio withdrawal during retirement in the presence of longevity risk Download PDFInfo
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- US8781937B2 US8781937B2 US13/602,996 US201213602996A US8781937B2 US 8781937 B2 US8781937 B2 US 8781937B2 US 201213602996 A US201213602996 A US 201213602996A US 8781937 B2 US8781937 B2 US 8781937B2
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- the present invention comprises a method and system for recommending the amount a retiree should withdraw (optimal withdrawal amount) from investment portfolio accounts comprised of financial assets and products, over a given period, for the purpose of financing retirement. More specifically, the present invention is a method and system for recommending the optimal withdrawal amount, for a given period, from a portfolio of assets or products used to finance retirement, taking into account the retiree's comprehensive level of risk aversion and any pension or other life annuity income of the retiree.
- the global population is ageing and the elderly are living longer than ever before.
- the National Vital Statics Report published by the National Center for Health Statistics, indicates that global human life expectancy increased from 59.7 years in 1930 to approximately 78.3 years by 2011. The increase can be attributed to improvements in infectious disease control, public health initiatives and medical innovations.
- the size of the elderly population both in absolute and relative terms, has been steadily increasing.
- the United States Census Bureau in a 2010 census brief titled “Age and Sex Composition: 2010” (issued in May 2011), reported that there were 40.2 million people aged 65 and older in the United States in 2010, representing roughly 13% of the U.S. population.
- the elderly population increased 15.1% from 2000 to 2010.
- the median age in the U.S. has increased from 29.5 years in 1960 to 37.2 in 2010.
- the utility maximizing level of spending is a function of the retiree's time and risk preferences as well as projections of future economic, financial and market conditions. It is well established in the financial economics literature—see for example Robert Merton (1990)—that the resulting optimal spending strategy, assuming constant relative risk aversion (CRRA), requires that the retiree invest in a constant risk portfolio and spend in proportion to the total value of his/her retirement assets. Since the retiree's wealth is dependent on market factors, spending rates are not constant.
- Another distinguishing feature of our methodology is the way it manages wealth drawdowns and, eventually, depletions over time. Whereas practitioners attempt to minimize the chance of the retiree's wealth being depleted prior to death, or some fixed time horizon, our methodology does not attempt to arbitrarily extend the so-called optimal wealth depletion time (WDT). In fact, in certain cases, where the retiree receives substantial pension or other annuity income, it is conceivable that the optimal spending trajectory results in an optimal wealth depletion time (WDT) that is prior to the date of death. In other words, depending on a retiree's level of longevity risk aversion and annuity income, it may be advisable for the retiree to deplete his or her portfolio accounts prior to death and live entirely off of pension annuity income. While the optimal wealth depletion time is obviously a personal preference, and highly dependent on one's attitude towards creating a legacy and financial bequest, this approach and methodology is lacking in the existing methodologies.
- the invention provides a system, method, and machine-readable medium for recommending an optimal withdrawal amount, for a given period, from a retiree's portfolio accounts, comprised of relatively risky and relatively safe assets, used to finance retirement.
- the user supplies information about the retiree's personal characteristics, including the retiree's age, health status, and gender. Details of the retiree's financial situation are also supplied, including the retiree's total liquid wealth, the current value of relatively risky and relatively safe assets, and any after-tax pension and other annuity income.
- Risk standard deviation of return
- return expected rate of return based on a lognormal or other random distribution
- or other measurable differentiating characteristics are retrieved for a portfolio comprised of relatively risky assets.
- a valuation rate is also retrieved.
- the retiree's level of overall risk aversion is used as a proxy for their specific level of longevity risk aversion, which is computed using a function of the expected rate of return for a portfolio of relatively risky assets, the valuation rate, the standard deviation of return for a portfolio of relatively risky assets, and the percentage of the retiree's total wealth comprised of relatively risky assets.
- the retiree's optimal wealth depletion time which is a function of the retiree's wealth, after-tax pension and other annuity income, the valuation rate, the retiree's level of longevity risk aversion, the retiree's probability of survival, and the retiree's actuarial discount rate, is solved over multiple iterations or using any one of an assortment of search methods.
- the actuarial discount rate is a function of the retiree's age, probability of survival, the valuation rate, the retiree's subjective discount rate, and a time variable.
- the retiree's optimal withdrawal amount is computed as a function of said retiree's wealth, after-tax pension and other annuity income, optimal wealth depletion time, the valuation rate and the actuarial discount rate.
- the resulting withdrawal recommendation is displayed or otherwise communicated to the user. In the preferred embodiment, it can be displayed as a single withdrawal from one portfolio. In alternative embodiments, it can be displayed as a collection of withdrawals from a multitude of portfolios, with each portfolio having a different regulatory or tax classification Systems and media are provided to carry out the above embodiments of the invention.
- FIG. 1A is a chart showing the expected age at death for females in the United States from 1933 to 2007;
- FIG. 1B is a chart showing the expected age at death for males in the United States from 1933 to 2007;
- FIG. 2A is a chart showing the standard deviation of remaining life for females in the United States from 1933 to 2007;
- FIG. 2B is a chart showing the standard deviation of remaining life for males in the United States from 1933 to 2007;
- FIG. 3 shows the optimal portfolio withdrawal amount (OWA) for retirees at different ages, levels of pension income, and levels of longevity risk aversion;
- FIG. 4 is a flowchart of an embodiment of the invention in which a basic search method is used to solve the optimal wealth depletion time
- FIG. 5 is a flowchart detail of FIG. 4 illustrating steps to be taken in calculating the present value of the retiree's pension and other annuity income;
- FIG. 6 is a flowchart detail of FIG. 4 illustrating one method for calculating the actuarial discount rate
- FIG. 7 is a schematic diagram of a system for carrying out the invention.
- FIG. 8 is a diagram illustrating the typical internal architecture of a server suitable for carrying out the invention.
- FIG. 7A is a schematic diagram of an alternative system for carrying out the invention.
- FIG. 8A is a diagram illustrating the typical internal architecture of a personal computer suitable for carrying out the invention.
- the investor's optimal withdrawal amount is influenced by longevity risk aversion—the “fear” of living longer than anticipated and possibly depleting one's financial resources.
- longevity risk aversion the “fear” of living longer than anticipated and possibly depleting one's financial resources.
- the illustrated embodiment describes the invention using two primary categories of assets: relatively risky financial assets and relatively safe financial assets.
- the invention accounts for any pension or other annuity income the retiree may receive.
- OWA( t ) OWA(0)*(EXP( k*t ))*( P ( t,x,L,M,B )) ⁇ (1/LRA) (1)
- k ( r ⁇ SDR)/LRA
- OWA(0) N 1/ D 1 (2)
- N 1 ( W+ (PAI(0)/ r ))*(EXP( r *WDT)) ⁇ (PAI(0)/ r )
- D 1 ( a (WDT, x,r,k,L,M,B ))*(EXP( r *WDT))
- OWA(WDT) ⁇ PAI(0) 0
- N 1 ( W+ (PAI(0)/ r ))*(EXP( r *WDT)) ⁇ (PAI(0)/ r )
- D 1 ( a (WDT, x,r,k,L,M,B ))*(EXP( r *WDT))
- N 2 (EXP( k *WDT))*( P ( t,x,L,M,B )) ⁇ (1/LRA)
- LRA is the retiree's level of longevity risk aversion.
- conditional probability of a person of age x surviving t years under the Gompertz Law of Mortality P(t,x,L,M,B)
- P(t,x,L,M,B) EXP(( ⁇ L*t )+(EXP(( x ⁇ M )/ B ))*(1 ⁇ EXP( t/B )) (5)
- P ( t,x,L,M,B ) EXP(( ⁇ L*t )+(EXP(( x ⁇ M )/ B ))*(1 ⁇ EXP( t/B )) (5)
- the mean (Mu) and standard deviation (Sigma) have been used to define the expected rate of return and risk, respectively.
- the use of mean and standard deviation in turn assumes the use of a standard lognormal distribution.
- investments in relatively risky assets have a random return X within a distribution defined by two moments: a standard mean and a statistical measure of variation.
- standard normal and other distributions that have more or less skewness and kurtosis.
- a standard normal distribution with no skew can be used.
- Johnson distribution is a set of statistical distributions parameterized by four variables.
- Other distributions can be found in William H. Greene, Econometric Analysis, Prentice Hall, 3 rd ed. (1997). Determining the Optimal Level of Withdrawals
- FIG. 4 is a flowchart presenting an overview of one embodiment of the present invention.
- the first step 400 is the supply of various inputs, including the retiree's age, gender, health status, the amount invested in relatively risky and relatively safe assets, and any periodic after-tax pension or other annuity income. According to the present embodiment, the user will supply each of these inputs.
- step 402 various capital markets inputs are retrieved from a database, including the valuation rate, the expected rate of return on a portfolio or index comprised of relatively risky assets, and the standard deviation of returns for said portfolio or index.
- the system computes the present value of the retiree's after-tax pension or other annuity income, PVAnnuity.
- FIG. 5 presents one method for computing PVAnnuity.
- the first step 500 is the retrieval of Gompertz mortality parameters (L, M and B) from a database.
- the next step 502 is to set the initial value of PVAnnuity equal to 0.
- the range of values for T is set. In the present embodiment, T cannot exceed 122 less the age of retiree since it is highly improbable that the retiree will live beyond 122 years of age.
- the final step 508 instructs the algorithm to proceed to the next value of T for each value of T within the established range.
- step 406 we compute the variable Alpha, which is the percentage of the retiree's assets invested in relatively risky assets.
- Wealth variable is the value of financial assets held by the retiree excluding any pension and annuity (or similar) assets.
- the variable Pie is computed by “annualizing” any monthly after-tax pension and income cash flow streams.
- WDT optimal wealth depletion time
- FIG. 6 contains a flowchart illustrating the process for calculating the value of APVX.
- the first step 600 is to retrieve the Gompertz Mortality Parameters (L, M and B).
- the initial value of APVX is set equal to 0.
- the value of T which, loosely speaking, means “time” is set equal to 0 in step 604 .
- APVT actuarial present value at time T
- C 1 EXP(( ⁇ L*T +EXP((Age ⁇ M )/ B )*(1 ⁇ EXP( T/B ))
- C 2 (1+ r ) ⁇ T
- the numerator is the conditional probability of surviving T years given the retiree's age, which has been parameterized on the basis of the Gompertz law of mortality (see equation (5)).
- the system determines whether the value of T exceeds the optimal wealth depletion time (WDT). Where this condition is not satisfied, the value of APVX is increased by the value of APVT (step 610 ), the value of T is increased by 1 (step 612 ) and steps 606 to 612 are repeated until the value of T exceeds the value of WDT at step 608 .
- the value of APVX is stored (step 614 ) and retrieved at step 420 (see FIG. 4 ).
- the actuarial present value (APVX) from time 0 to WDT is the sum of the individual actuarial present values computed for each value of T ranging from 0 to WDT.
- the flowchart in FIG. 6 presents one method for integrating these amounts.
- N 1 (Wealth ⁇ A 0)* A 1 ⁇ A 0
- D 1 A 5*
- a 2 EXP( k *WDT)
- a 4 POW( ⁇ 3,1/gamma)
- a 0 Pie/ r
- a 1 EXP( r *WDT)
- a 3 EXP(( ⁇ L *WDT+EXP((Age ⁇ M )/ B )*(1 ⁇ EXP(WDT/ B ))
- step 426 assesses whether WDTF is greater than or less than 0. If WDTF>0, the value of WDT is increased by 1/12 (step 428 ), the value of APVX is recalculated (transition from step 428 to step 420 ) and the value of WDTF is recalculated (step 422 ). If WDTF ⁇ 0, the value of WDT is decreased by 1/12 (step 430 ), the value of APVX is recalculated (transition from step 430 to step 420 ) and the value of WDTF is recalculated (step 422 ).
- WDT wealth depletion time
- FIG. 8 shows a high-level schematic internal architecture of server 102 .
- the server 102 's main logic is embodied by a general-purpose, programmable microprocessor(s) 104 , which in conventional practice will have an on-board memory cache (not shown) and which may be associated with one or more mathematical or other special-purpose coprocessors.
- the processing logic represented by processor 104 is connected by a bus structure 106 to the various other components of server 102 .
- To execute the algorithm shown in FIGS. 4 , 5 and 6 where that algorithm is written in a programming language, at least a dual core processor is recommended.
- bus 106 The schematic representation of bus 106 is illustrated in FIG. 8 as a simple and unitary structure; however, in conventional practice, as is known by anyone skilled in the art, there are usually several buses (see, for example, bus 114 ) and communication pathways that operate at different speeds and have multiple purposes.
- Server 102 will also have random access memory (RAM) units 108 connected to bus 106 .
- RAM 108 DRAM, SDRAM, or other known types
- RAM 108 typically has loaded into it the operating system of the server 102 and executable instructions for one or more special applications designed to carry out the invention.
- a computer program suitable for carrying out the invention will be stored on a mass storage device 112 , such as an optical disk, magnetic drive, or a solid state drive.
- the data for both relatively safe and relatively risky assets, used to compute the retiree's level of longevity risk aversion, and Gompertz mortality parameters (L, M, and B), will typically reside on a separate database server 126 that can be accessed remotely through the Internet 130 ; in the alternative, the data could exist as a database on device 112 .
- Bus 106 connects mass storage device 112 to RAM 108 .
- an operator using computer 128 can manage information stored on database 126 through the medium of the Internet 130 , as shown in FIG. 7 .
- the server 102 is connected to various peripheral devices used to communicate with an operator, such as display 116 , keyboard 118 , and mouse 120 .
- the server 102 also uses a communications device 122 such as on-board network connectivity or an attached network card to communicate to other computers and equipment. This allows the operator to input programming instructions directly onto server 102 , among other things.
- an operator using computer 128 can upload programming instructions to server 102 through the medium of the Internet 130 .
- the optimal withdrawal amount (OWA) may be used to periodically recommend optimal levels of withdrawals from a retiree's portfolio accounts.
- the server 102 calculates the optimal withdrawal amount (OWA) based in part on data obtained from database 126 .
- Other inputs used in calculating the optimal withdrawal amount (OWA) can be input by a retiree from a remote computer 132 and then transmitted to server 102 through the medium of the Internet 130 .
- These data may include, for example, the retiree's wealth, current asset allocations to relatively risky and relatively safe assets, age, health status, and any pension and annuity income amounts.
- FIG. 7A A representative system for carrying out the invention, under this alternative scenario, is illustrated in FIG. 7A .
- a system for maximizing the utility of consumption during retirement 100 may be assembled around a programmed, general-purpose computer 102 having personal computer (“PC”) architecture.
- PC personal computer
- FIG. 8A a highly schematic internal architecture of computer 102 is shown.
- computer 102 's main logic is embodied by a general-purpose, programmable microprocessor 104 .
- the microprocessor 104 will, in conventional practice, have an on-board memory cache (not shown) and may be associated with one or more mathematics or other special-purpose coprocessors (not shown).
- the processing logic represented by microprocessor 104 is connected via bus structure 106 to the other components of the computer 102 .
- bus 106 appears in FIG. 8A as a simple and unitary structure; however, in practice, as is known to those in the art, there are usually several buses and communication pathways 106 operating at different speeds and having different purposes. Furthermore, bus 106 may be segmented and controlled by bus controllers, as is also known in the art.
- Computer 102 will also have a random access memory (RAM) unit, or units, 108 connected to bus 106 .
- RAM 108 (which may be DRAM, SDRAM or other known types) typically has loaded into it the operating system of the computer 102 and executable instructions for one or more special applications designed to carry out the invention.
- a computer program suitable for carrying out the invention will be stored on a mass storage device 112 , such as an optical disk or magnetic hard drive.
- Bus 106 connects mass storage device 112 to RAM 108 .
- the asset data used to compute the retiree's level of longevity risk aversion, and Gompertz mortality parameters (L, M, and B), will typically exist as a database on device 112 but it could also reside on a separate database server and be accessed remotely through a network.
- the computer 102 is connected to peripheral devices used to communicate with an operator, such as display 114 , keyboard 116 , mouse 118 , and printer 120 .
- the computer 102 also uses a communications device 122 such as a modem or a network card to communicate to other computers and equipment.
- the optimal withdrawal amount (OWA) may be used to periodically recommend optimal levels of withdrawals from a retiree's portfolio accounts.
- computer 102 calculates the optimal withdrawal amount (OWA) based, in part, on data obtained from database 112 .
- Other inputs used in calculating the optimal withdrawal amount (OWA) can be input by the user from the keyboard 116 and mouse 118 .
- the result is displayed on display 114 .
- the end user would install the application on a personal hand-held device such as a Tablet computer or a so-called Smart Phone device.
- the device's main logic is embodied by a general-purpose, programmable processor.
- the processing logic represented by the processor is connected to the other components such as a screen/panel, memory (both volatile and non-volatile), network communications components, and other peripheral devices.
- the tablet or smart phone device will also have a memory which has loaded onto it the operating system of the tablet or smart phone device and executable instructions for one or more special applications designed to carry out the invention.
- a computer program suitable for carrying out the invention will be stored on a non-volatile memory such as a flash device.
- the asset data used to compute the retiree's level of longevity risk aversion, and Gompertz mortality parameters (L, M, and B), will typically exist as a database on the device but it could also reside on a separate database server and be accessed remotely through the device's communications components.
- the optimal withdrawal amount may be used to periodically recommend optimal levels of withdrawals from a retiree's portfolio at the beginning of the period.
- the tablet or smart phone device's processor calculates the optimal withdrawal amount (OWA) based, in part, on data obtained from the database. Other inputs used in calculating the optimal withdrawal amount (OWA) can be input by the user from the keypad. Once the device has calculated the optimal withdrawal amount (OWA), the result is displayed on the device's screen.
- Table 1 in Appendix B, shows the optimal withdrawal amount for a retiree under different age and pension assumptions.
- the retiree has $1,000,000 in assets in his portfolio of which $350,000 (35%) is invested in relatively risky assets and the remaining $650,000 (65%) is invested in relatively safe assets.
- the portfolio is assumed to grow at a rate of 3.9%.
- the retiree's level of longevity risk aversion risk aversion (gamma) is assumed to be 2.14 units. All of this is assuming the real expected rate of return on a portfolio of relatively risky assets is 5.5%, the standard deviation of return on that portfolio is 20%, and the valuation rate is 2.5%.
- the table also shows that our retiree consumes more from his portfolio at higher levels of pension income. For example, at age 70 with a pension of $50,000 per year, our retiree would withdraw $74,070 from his portfolio. In contrast, with a pension of only $20,000 per year, the retiree would consume $66,900 from his portfolio at age 70. In essence, the pension serves as a buffer and enables the retiree to spend more from discretionary wealth. The retiree is not as concerned with longevity risk because he has pension income to fall back on should that risk (i.e. living longer than anticipated) materialize.
- Table 2 in Appendix A shows the optimal withdrawal amount for four retirees with different levels of longevity risk aversion (gamma values).
- the retirees are identical in every other respect. Each has a portfolio of assets valued at $1,000,000 at age 65 and each retiree's portfolio is growing at 4.5%. Assuming identical portfolio growth rates for retirees with different asset allocations is highly unrealistic; that said, this treatment is necessary to isolate the effect of changes in longevity risk aversion on optimal spending levels.
- Gamma values are computed assuming the expected rate of return on a portfolio comprised solely of relatively risky assets is 5.5% and the valuation rate is 1.5%. Furthermore, it is assumed that the retiree has zero pension or other annuity income.
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Abstract
Description
OWA(t)=OWA(0)*(EXP(k*t))*(P(t,x,L,M,B))^(1/LRA) (1)
where k=(r−SDR)/LRA
In the above expressions:
-
- OWA(0)=the initial optimal withdrawal amount (withdrawals at t=0)
- P(t,x,L,M,B)=the objective probability of a person of age x surviving to time t parameterized using Gompertz mortality parameters L, M, and B
- LRA=the retiree's level of longevity risk aversion
- r=the valuation rate
- SDR=the retiree's subjective discount rate
OWA(0)=N1/D1 (2)
Where:
N1=(W+(PAI(0)/r))*(EXP(r*WDT))−(PAI(0)/r)
D1=(a(WDT,x,r,k,L,M,B))*(EXP(r*WDT))
In the above expression:
-
- W denotes the wealth level
- PAI(0) denotes the initial level of periodic after-tax pension and other annuity income
- r denotes the valuation rate
- WDT denotes the optimal wealth depletion time
- a(WDT,x,r,k,L,M,B) denotes the retirement age “price” of a life-contingent pension annuity that pays a real $1 a year until death or time WDT, whichever comes first
OWA(WDT)−PAI(0)=0
In other words, if an optimal wealth depletion time exists, then for spending to remain smooth at that point, which is part of the foundation of life-cycle theory, it must converge to PAI(0). More simply, the retiree's wealth will have been depleted at the point where the only remaining source of funding in retirement is pension and/or annuity income. Mathematically, the optimal wealth depletion time, WDT, satisfies the following equation:
(N1/D1)*N2=PAI(0) (3)
Where:
N1=(W+(PAI(0)/r))*(EXP(r*WDT))−(PAI(0)/r)
D1=(a(WDT,x,r,k,L,M,B))*(EXP(r*WDT))
N2=(EXP(k*WDT))*(P(t,x,L,M,B))^(1/LRA)
LRA=N1/D1 (4)
Where:
N1=(Mu−r)/Sigma
D1=Alpha
In the above expression:
-
- Mu denotes expected rate of return on a portfolio of relatively risky assets
- r denotes the valuation rate
- Alpha denotes the percentage of the retiree's total wealth allocated to relatively risky assets
- Sigma denotes the standard deviation of returns for a relatively risky portfolio of assets
P(t,x,L,M,B)=EXP((−L*t)+(EXP((x−M)/B))*(1−EXP(t/B)) (5)
In the above expression:
-
- L denotes the hazard rate due to accidents
- M denotes the modal value of life (e.g., 80 years)
- B denotes the dispersion coefficient (e.g., 10 years) of the future lifetime random variable
The Gompertz-Makeham mortality model parameters denoted by letters L, M, and B are calculated by fitting a Gompertz-Makeham survival probability curve to the mortality table that is available from known sources, such as the Society of Actuaries. Since the mortality tables contain death probabilities, the survival probability is calculated by subtracting from 1.0. Mathematically, many techniques are available for locating the parameters of the Gompertz-Makeham distribution. Particularly, in the preferred embodiment, we use the Gauss-Newton method to calculate the parameters.
The present invention incorporates the following decision factors:
LogNormalPDF(x;Mu,Sigma)=(1/(((2*Pi)^(½))*Sigma*x))*EXP(−((LN(x)−Mu)/Sigma)^2)/2),x>=0
N(x;Mu,Sigma)=(1/(((2*Pi)^(½))*Sigma))*EXP((−((x−Mu)/Sigma)^2)/2)
X=c+d*(Phi^(−1))*((Z−a)/b) Z=a+b*Phi*((X−c)/d)
Other distributions can be found in William H. Greene, Econometric Analysis, Prentice Hall, 3rd ed. (1997).
Determining the Optimal Level of Withdrawals
PVAnnuity=PVAnnuity+(C0*C1)/C2
Where:
C0=Pension+Annuity
C1=EXP((−L*T+EXP((Age−M/B)*(1−EXP(T/B))
C2=(1+r)^T
More simply, PVAnnuity is the present value of any periodic after-tax pension and/or annuity cash inflows received by the retiree, adjusted for the probability of the retiree being alive to receive the cash flow.
Alpha=Risky Assets/(RiskyAssets+SafeAssets+PVAnnuity)
gamma=(Mu−r)/(Alpha*Sigma)
In the event that alpha is equal to zero, we proceed to step 412 wherein gamma is assigned a value of 100. Note that gamma can be assigned any value sufficiently large such that 1/gamma becomes a very small number; this has the effect of reducing the optimal withdrawal amount for a retiree who is highly risk averse.
Wealth=Risky Assets+SafeAssets
Note that the Wealth variable is the value of financial assets held by the retiree excluding any pension and annuity (or similar) assets.
Pie=12*(Pension+Annuity)
APVT=C1/C2
Where:
C1=EXP((−L*T+EXP((Age−M)/B)*(1−EXP(T/B))
C2=(1+r)^T
Note that the numerator is the conditional probability of surviving T years given the retiree's age, which has been parameterized on the basis of the Gompertz law of mortality (see equation (5)).
WDTF=N1/D1*A2*A4−Pie
Where:
N1=(Wealth−A0)*A1−A0
D1=A5*A1
A2=EXP(k*WDT)
A4=POW(^3,1/gamma)
A0=Pie/r
A1=EXP(r*WDT)
A3=EXP((−L*WDT+EXP((Age−M)/B)*(1−EXP(WDT/B))
A5=APVX
Step 424 verifies whether this condition (WDTF≈0) is satisfied. If the condition is not satisfied, we proceed to step 426 which assesses whether WDTF is greater than or less than 0. If WDTF>0, the value of WDT is increased by 1/12 (step 428), the value of APVX is recalculated (transition from
OWA=N2/D2
Where:
N2=(Wealth+B0)*B1−B0
D2=B2*B1
B0=Pie/r
B1=EXP(r*WDT)
B2=APVX
In the
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US20140236637A1 (en) * | 2013-02-21 | 2014-08-21 | Antonio Limjuco Tan-Torres, JR. | Horizon-based settlement of defined benefit pension plan liabilities |
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